Abstract

Let 𝐺 be a graph with vertex set 𝑉=(𝑣1,𝑣2,…,𝑣𝑛). Let 𝛿(𝑣𝑖) be the degree of the vertex 𝑣𝑖∈𝑉. If the vertices 𝑣𝑖1,𝑣𝑖2,…,𝑣𝑖ℎ+1 form a path of length ℎ≥1 in the graph 𝐺, then the ℎth order Randić index 𝑅ℎ of 𝐺 is defined as the sum of the terms 1/𝛿(𝑣𝑖1)𝛿(𝑣𝑖2)⋯𝛿(𝑣𝑖ℎ+1) over all paths of length ℎ contained (as subgraphs) in 𝐺. Lower and upper bounds for 𝑅ℎ, in terms of the vertex degree sequence of its factors, are obtained for corona product graphs. Moreover, closed formulas are obtained when the factors are regular graphs.

1. Introduction

In this work we consider simple graphs 𝐺=(𝑉,𝐸) with 𝑛 vertices and 𝑚 edges. Let 𝑉=(𝑣1,𝑣2,…,𝑣𝑛) be the vertex set of 𝐺. For every vertex 𝑣𝑖∈𝑉, 𝛿(𝑣𝑖) represents the degree of the vertex 𝑣𝑖 in 𝐺. The maximum and minimum degree of the vertices of 𝐺 will be denoted by Δ and 𝛿, respectively.

The Randić index 𝑅1(𝐺) of a graph 𝐺 was introduced in 1975 [1] and defined as 𝑅1(𝐺)=𝑣𝑖𝑣𝑗∈𝐸1𝛿𝑣𝑖𝛿𝑣𝑗.(1.1)
This graph invariant, sometimes referred to as connectivity index, has been successfully related to a variety of physical, chemical, and pharmacological properties of organic molecules, and it has became into one of the most popular molecular-structure descriptors. After the publication of the first paper [1], mathematical properties of 𝑅1 were extensively studied, see [2–6] and the references cited therein.

The higher-order Randić indices are also of interest in chemical graph theory. For ℎ≥1, the ℎth order Randić index 𝑅ℎ(𝐺) of a graph 𝐺 is defined as 𝑅ℎ(𝐺)=𝑣𝑖1𝑣𝑖2⋯𝑣𝑖ℎ+1∈𝒫ℎ(𝐺)1𝛿𝑣𝑖1𝛿𝑣𝑖2𝑣⋯𝛿𝑖ℎ+1,(1.2)
where 𝒫ℎ(𝐺) denotes the set of paths of length ℎ contained (as subgraphs) in 𝐺. Of the higher-order Randić indices the most frequently applied is 𝑅2 [7–10]. Estimations of the higher-order Randić index of regular graphs and semiregular bipartite graphs are given in [10]. In this paper we are interested in studying the higher-order Randić index, 𝑅ℎ, for corona product graphs. Roughly speaking, we study the cases ℎ=1, ℎ=2 for arbitrary graphs and the case ℎ≥3 when the second factor of the corona product is an empty graph. As an example of a chemical compound whose graph is obtained as a corona product graph we consider the Cycloalkanes with a single ring, whose chemical formula is 𝐶𝑘𝐻2𝑘, and whose molecular graph can be expressed as 𝐶𝑘⊙𝑁2, where 𝐶𝑘 is the cycle graph of order 𝑘 and 𝑁2 is the empty graph of order two. We recall that, given two graphs 𝐺 and 𝐻 of order 𝑛1 and 𝑛2, respectively, the corona product 𝐺⊙𝐻 is defined as the graph obtained from 𝐺 and 𝐻 by taking one copy of 𝐺 and 𝑛1 copies of 𝐻 and then joining by an edge each vertex of the 𝑖th copy of 𝐻 with the 𝑖th vertex of 𝐺.

Proof. Let 𝑉1={𝑣1,𝑣2,…,𝑣𝑛1} and 𝑉2={𝑢1,𝑢2,…,𝑢𝑛2} be the set of vertices of 𝐺1 and 𝐺2, respectively. Given a vertex 𝑣∈𝑉𝑖, we denote by 𝑁𝐺𝑖(𝑣) the set of neighbors that 𝑣 has in 𝐺𝑖. The paths of length two in 𝐺1⊙𝐺2 are obtained as follows: (i)paths 𝑢𝑖𝑣𝑗𝑢𝑘, 𝑖≠𝑘, where 𝑢𝑖,𝑢𝑘∈𝑉2 and 𝑣𝑗∈𝑉1, (ii)paths 𝑢𝑖𝑣𝑗𝑣𝑘, 𝑗≠𝑘, where 𝑢𝑖∈𝑉2 and 𝑣𝑗𝑣𝑘∈𝑉1, (iii)paths 𝑣𝑖𝑢𝑗𝑢𝑘, 𝑗≠𝑘, where 𝑣𝑖∈𝑉1 and 𝑢𝑗,𝑢𝑘∈𝑉2, (iv)paths of length two belonging to 𝐺1, (v)paths of length two belonging to the 𝑛1 copies of 𝐺2.
So, we have 𝑅2(𝐺1⊙𝐺2∑)=5𝑖=1𝑄𝑖, where
𝑄1=𝑣𝑗∈𝑉1;𝑢𝑖,𝑢𝑘∈𝑉21𝛿𝑢𝑖𝛿𝑣+1𝑗+𝑛2𝛿𝑢𝑘=+1𝑛1𝑗=11𝛿𝑣𝑗+𝑛2⋅𝑛2−1𝑛𝑖=12𝑙=𝑖+11𝛿𝑢𝑖𝛿𝑢+1𝑙≥𝑛+11𝑛2𝑛2−12Δ2√+1Δ1+𝑛2(2.6)
corresponds to the paths type (i),
𝑄2=𝑢𝑖∈𝑉2;𝑣𝑗,𝑣𝑘∈𝑉11𝛿𝑢𝑖𝛿𝑣+1𝑗+𝑛2𝛿𝑣𝑘+𝑛2=𝑛2𝑖=11𝛿𝑢𝑖⋅+1𝑛1𝑗=1𝑣𝑙∈𝑁𝐺1(𝑣𝑗)1𝛿𝑣𝑗+𝑛2𝛿𝑣𝑙+𝑛2≥2𝑚1𝑛2Δ1+𝑛2√Δ2+1(2.7)
corresponds to the paths type (ii),
𝑄3=𝑣𝑖∈𝑉1;𝑢𝑗,𝑢𝑘∈𝑉21𝛿𝑣𝑖+𝑛2𝛿𝑢𝑗𝛿𝑢+1𝑘=+1𝑛1𝑖=11𝛿𝑣𝑖+𝑛2⋅𝑛2𝑗=1𝑢𝑙∈𝑁𝐺2(𝑢𝑗)1𝛿𝑢𝑗𝛿𝑢+1𝑙≥+12𝑛1𝑚2Δ2√+1Δ1+𝑛2(2.8)
corresponds to the paths type (iii),
𝑄4=𝑣𝑖𝑣𝑗𝑣𝑘𝐺∈𝒫11𝛿𝑣𝑖+𝑛2𝛿𝑣𝑗+𝑛2𝛿𝑣𝑘+𝑛2≥12Δ1+𝑛2𝛿𝑣𝑖≥2𝛿𝑣𝑖𝛿𝑣𝑖−1𝛿𝑣𝑖+𝑛2(2.9)
corresponds to the paths type (iv), and
𝑄5=𝑢𝑖𝑢𝑗𝑢𝑘𝐺∈𝒫21𝛿𝑢𝑖𝛿𝑢+1𝑗𝛿𝑢+1𝑘≥1+12Δ2+1𝛿𝑢𝑖≥2𝛿𝑢𝑖𝛿𝑢𝑖−1𝛿𝑢𝑖+1(2.10)
corresponds to the paths type (v). Thus, the lower bound follows. The upper bound is obtained by analogy.

Corollary 2.4. For 𝑖∈{1,2}, let 𝐺𝑖 be a 𝛿𝑖-regular graph of order 𝑛𝑖. Then,
𝑅2𝐺1⊙𝐺2=𝑛1𝑛2𝛿2√+1𝛿1+𝑛2𝑛2−12+𝛿2+𝑛1𝛿12𝛿1+𝑛22𝑛2√𝛿2+𝛿+11−1√𝛿1+𝑛2+𝑛2𝛿2𝛿2−12𝛿2√+1𝛿2.+1(2.11)

The girth of a graph is the size of its smallest cycle. For instance, the molecular graphs of benzenoid hydrocarbons have girth 6. The molecular graphs of biphenylene and azulene have girth 4 and 5, respectively [11].

The following result, and its proof, was implicitly obtained in the proof of Theorem 1 of [10]. By completeness, here we present it as a separate result.

Lemma 2.5. Let 𝐺=(𝑉,𝐸) be a graph with girth 𝑔(𝐺). If 𝛿≥2 and 𝑔(𝐺)>ℎ, then the number of paths of length ℎ in 𝐺 is bounded by
(𝛿−1)ℎ−22𝑢∈𝑉||𝒫𝛿(𝑢)(𝛿(𝑢)−1)≤ℎ||≤(𝐺)(Δ−1)ℎ−22𝑢∈𝑉𝛿(𝑢)(𝛿(𝑢)−1).(2.12)

Proof. Since 𝛿≥2, for every 𝑣∈𝑉, the number of paths of length 2 in 𝐺 of the form 𝑣𝑖𝑣𝑣𝑗 is 𝛿(𝑣)(𝛿(𝑣)−1)/2. Therefore, the result follows for ℎ=2.Suppose now that ℎ≥3. Given a vertex 𝑢∈𝑉, let 𝒫ℎ(𝑢) be the set of paths of length ℎ whose second vertex is 𝑢, that is, paths of the form 𝑢1𝑢𝑢2⋯𝑢ℎ. We denote by 𝑁(𝑣) the set of neighbors of an arbitrary vertex 𝑣∈𝑉. Note that the degree of 𝑣 is 𝛿(𝑣)=|𝑁(𝑣)|. If 𝛿≥2, then for every 𝑣∈𝑉 and 𝑤∈𝑁(𝑣) we have 𝑁(𝑤)⧵{𝑣}≠∅. So, for every 𝑢∈𝑉, there exists a vertex sequence 𝑢1𝑢𝑢2⋯𝑢ℎ such that 𝑢1,𝑢2∈𝑁(𝑢), 𝑢3∈𝑁(𝑢2)⧵{𝑢}, 𝑢4∈𝑁(𝑢3)⧵{𝑢2},…,and𝑢ℎ∈𝑁(𝑢ℎ−1)⧵{𝑢ℎ−2}. If 𝑔(𝐺)>ℎ, then the sequence 𝑢1𝑢𝑢2⋯𝑢ℎ is a path. Conversely, every path of length ℎ whose second vertex is 𝑢 can be constructed as above. Hence, the number of paths of length ℎ whose second vertex is 𝑢 is bounded by
||𝒫ℎ(||𝑢)≥min𝑢1𝑢𝑢2⋯𝑢ℎ∈𝒫ℎ(𝑢)𝛿(𝑢)(𝛿(𝑢)−1)ℎ−1𝑗=2𝛿𝑢𝑗−1≥𝛿(𝑢)(𝛿(𝑢)−1)(𝛿−1)ℎ−2,||𝒫ℎ||(𝑢)≤max𝑢1𝑢𝑢2⋯𝑢ℎ∈𝒫ℎ(𝑢)𝛿(𝑢)(𝛿(𝑢)−1)ℎ−1𝑗=2𝛿𝑢𝑗−1≤𝛿(𝑢)(𝛿(𝑢)−1)(Δ−1)ℎ−2.(2.13)
Thus, the result follows.

Now 𝑁𝑘 denotes the empty graph of order 𝑘.

Theorem 2.6. Let 𝐺=(𝑉,𝐸) be a graph with girth 𝑔(𝐺), minimum degree 𝛿, and maximum degree Δ. If 𝛿≥2 and 𝑔(𝐺)>ℎ≥3, then
𝑅ℎ𝐺⊙𝑁𝑘≤Δ−12√𝛿+𝑘+𝑘(Δ−1)ℎ−3(𝛿+𝑘)ℎ/2𝑢∈𝑉𝑅𝛿(𝑢)(𝛿(𝑢)−1),ℎ𝐺⊙𝑁𝑘≥𝛿−12√Δ+𝑘+𝑘(𝛿−1)ℎ−3(Δ+𝑘)ℎ/2𝑢∈𝑉𝛿(𝑢)(𝛿(𝑢)−1).(2.14)

Proof. The paths of length ℎ in 𝐺 contribute to Rℎ(𝐺⊙𝑁𝑘) in
𝑣𝑖1𝑣𝑖2⋯𝑣𝑖ℎ+1∈𝒫ℎ(𝐺)1∏ℎ+1𝑙=1𝛿𝑣𝑖𝑙+𝑘.(2.15)
Moreover, each path of length ℎ−1 in 𝐺 leads to 2𝑘 paths of length ℎ in 𝐺⊙𝑁𝑘; thus, the paths of length ℎ−1 in 𝐺 contribute to 𝑅ℎ(𝐺⊙𝑁𝑘) in
𝑣𝑖1𝑣𝑖2⋯𝑣𝑖ℎ∈𝒫ℎ−1(𝐺)2𝑘∏ℎ𝑙=1𝛿𝑣𝑖𝑙+𝑘.(2.16)
Hence,
𝑅ℎ𝐺⊙𝑁𝑘=𝑣𝑖1𝑣𝑖2⋯𝑣𝑖ℎ+1∈𝒫ℎ(𝐺)1∏ℎ+1𝑙=1𝛿𝑣𝑖𝑙++𝑘𝑣𝑖1𝑣𝑖2⋯𝑣𝑖ℎ∈𝒫ℎ−1(𝐺)2𝑘∏ℎ𝑙=1𝛿𝑣𝑖𝑙≤||𝒫+𝑘ℎ(||𝐺)√(𝛿+𝑘)ℎ+1||𝒫+2𝑘ℎ−1(||𝐺)√(𝛿+𝑘)ℎ.(2.17)
By Lemma 2.5 we obtain the upper bound and the lower bound is obtained by analogy.

Corollary 2.7. Let 𝐺=(𝑉,𝐸) be a 𝛿-regular graph of order 𝑛 and girth 𝑔(𝐺). If 𝛿≥2 and 𝑔(𝐺)>ℎ≥3, then
𝑅ℎ𝐺⊙𝑁𝑘=𝛿−12√𝛿+𝑘+𝑘𝑛𝛿(𝛿−1)ℎ−2(𝛿+𝑘)ℎ/2.(2.18)

Acknowledgment

This work was partly supported by the Spanish Government through projects TSI2007-65406-C03-01 “E-AEGIS” and CONSOLIDER INGENIO 2010 CSD2007-00004 “ARES.”